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Deep Convolutional Neural Networks on Cartoon Functions

by P. Grohs and T. Wiatowski and H. Helmut Boelcskei

(Report number 2016-25)

Abstract
Wiatowski and B\"olcskei, 2015, proved that deformation stability and vertical translation invariance of deep convolutional neural network-based feature extractors are guaranteed by the network structure per se rather than the specific convolution kernels and non-linearities. While the translation invariance result applies to square-integrable functions, the deformation stability bound holds for band-limited functions only. Many signals of practical relevance (such as natural images) exhibit, however, sharp and curved discontinuities and are hence not band-limited. The main contribution of this paper is a deformation stability result that takes these structural properties into account. Specifically, we establish deformation stability bounds for the class of cartoon functions introduced by Donoho, 2001.

Keywords: Convolutional Networks, Cartoon Functions, Deformation Stability

@Techreport{GWH16_662,
  author = {P. Grohs and T. Wiatowski and H. Helmut Boelcskei},
  title = {Deep Convolutional Neural Networks on Cartoon Functions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-25},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-25.pdf },
  year = {2016}
}

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